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looking at the Hyperbolic Exercises- page 90 of the text, q1 tells us we have a H-plane with two points, A and B (in addition to the center-point O) and that all these points rest on the same line.

We are asked to construct the H-line passing through A and B and to explain why this H-line is unique.

Is this simply a line seg thru A, O, and B (thus bisecting the H circle) and so it is unique because it is the only line seg (that first kind of H-line) that can pass through all 3 points?

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Will we need to know how to construct Julia Sets on the final exam, or just how to iterate more basic fractals like in the mid-term?

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Regarding the question about hyperbolic geometry: you are right. In fact there is only one H-line passing through any two distinct points.

Regarding fractals: You do not need to know Julia sets in the final. A problem involving iterations of fractals (of the type in the midterm) is an option.

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In the Fall 2003 Final Exam, we are asked (question 6a) to construct a hyperbolic line perpendicular to a given hyperbolic line (arc). In class we saw how to construct parallel hyperbolic lines, not perpendicular ones. Is a line (type 1 or type 2) perpendicular to the given arc if it touches it?

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In class we covered right angles between hyperbolic lines (or lines and circles). That is sufficient. That particular question is very easy, and the construction can be done in less than 5 seconds. The answer to your last question is negative: if a line touches a circle, then the angle between the two is 0.

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I am a little confused about the difference between homotopy in 2d versus homotopy in 3d. We covered it a little bit in class but the textbook does not mention it (as far as I can see), I was just wondering if you could clarify the differences between the two a little bit?

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In case of homotopy within 2D you are confined within the drawing plane, while homotopy within 3D allows you to get out of that plane and do things in 3 dimensions. For example, a point within a circle can be taken out of that circle (without cutting or gluing) in 3D but not in 2D. (That is, a planar object consisting of a circle and a point is homotopic in 3D to the object in the same plane consisting of a circle and a point out of it. However, these two objects are not homotopic within 2D i.e., within the plane).

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Euler Characeristic and 2-manifolds: just want to verify that the Euler Char of a double Torus is actually -2 ?

e(x)= 2-2g(x), and for double torus the genus is 2 (right?) so:

e(x) = 2-2(2) or e(x) = 2-4 = -2?

Anyone? My notes, somehow, look like a crazy person got to them...

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Right.

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referring to the final exam from spring 2003, question 7 a) I am confused as to how we would construct a line parallel to the given h-line?

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rephrasing the question posted above, I realize that 7 a) actually refers to a perpendicular line (not parallel). To construct a perpendicular to line L as asked (passing through point A) would one just draw a type one h-line which passes through the center point as well as point A?

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No - not so simple in this case. You can find this example solved in the notes in a more general setting (where the two h-lines intersect at any given angle). Here is a solution for this question, in a nutshell: find the center-line of A, and the tangent to the given h-line at A (both constructions were done in class); where they intersect is the center of the circle through A giving the desired h-line.

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and this creates an h-line of type 2?....sorry I am still rather confused by this construction, I can not seem to locate the general solving steps in my notes.

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Yes. The steps are given above; here it is again.

Step 1. Find the center-line for A (this has been done in both the math and in the art part of the course).

Step 2. Find the tangent at A to the given hyperbolic line (this has also been done in class: join the center of the circle defining the given h-line with A and construct the perpendicular at A).

Step 3: Make a circle centered at the intersection B of the two lines constructed in Steps 1 and 2, and of radius BA. The part of that circle within the hyperbolic plane is the wanted h-line.

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Hey, maybe you can answer this quick question because i cant seem to find the answer anywhere. How do you find the tangent of A to get A'?

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It is on page 146 in the textbook.